Chapter 23, Problem 9QP

Binomial Model In the previous problem, assume that the exercise style on the option is American rather than European. What is the price of the option now? (Hint: How will you find the value of the option if it can be exercised early? When would you exercise the option early?)

Summary Introduction

To determine: The price of put option.

Binomial Model:

Binomial model is the option of the modal in which price of the modal move according to the market situation. If the price of the option rises, it means that economic scenario is good and vice versa.

Explanation of Solution

Given,

Current stock price is $82.

Standard deviation on stock return is 70%.

Strike price of the option is $90.

Risk free rate is 5%.

The price of the put option will be three by using one-month steps.

Calculate the price of the put value stock,

$\text{Put}\text{value}=\frac{\left[{\text{P}}_{\text{Rise}}\left({\text{C}}_0\right)+{\text{P}}_{\text{Fall}}\left({\text{C}}_0\right)\right]}{1+\text{Risk}\text{free}\text{rate}}$

Where,

• ${\text{P}}_{\text{Rise}}$ is the rise in the price.
• ${\text{C}}_0$ is the put value of the option.
• ${\text{P}}_{\text{Fall}}$ is the fall in price.

Substitute 0.46 for the ${\text{P}}_{\text{Rise}}$, $0 for the put value option, 0.54 for the ${\text{P}}_{\text{Fall}}$, and 0.42% for the risk free rate.

$\begin{array}{c}\text{Put}\text{value}\left(\text{1}\right)=\frac{\left[0.46\left(\$0\right)+0.54\left(\$7.01\right)\right]}{\left(1+0.0042\right)}\\ =\frac{\$3.7854}{1.0042}\\ =\$3.7695\end{array}$

Substitute, 0.46 for the ${\text{P}}_{\text{Rise}}$, $7.01 for the put value option ${\text{C}}_{\text{1}}$ , 0.54 for the ${\text{P}}_{\text{Fall}}$, and 0.42% for the risk free rate and 55.401 for the put value of the $C_2$,

$\begin{array}{c}\text{Put}\text{value}\left(2\right)=\frac{\left[0.46\left(\$7.01\right)+0.54\left(\$55.401\right)\right]}{\left(1+0.0042\right)}\\ =\frac{\$33.14114}{1.0042}\\ =\$33.0025\end{array}$

Substitute 0.46 for the ${\text{P}}_{\text{Rise}}$, $266.54 for the put value 1 and $2333.88 for the put value 2 and 0.42% for the risk free rate.

$\begin{array}{c}\text{Put}\text{value}\left(3\right)=\frac{\left[0.46\left(\$266.54\right)+0.54\left(\$2,333.88\right)\right]}{\left(1+0.0042\right)}\\ =\frac{\$122.6084+\$1260.2952}{1.0042}\\ =\frac{\$1382.9036}{1.0042}\\ =\$1,377.1197\end{array}$

Working notes:

To compute the value of the stock by binomial modal, first find the value of the U and D,

Calculate the value of the U,

$\text{U}={\text{e}}^{\frac{\sigma }{\sqrt{\text{n}}}}$

Where,

• Percentage of the price rice up is U.
• Exponential factor is e (2.718)
• Standard deviation is $\sigma$.
• Number of the month is n

$\begin{array}{c}\text{U}={2.718}^{\frac{0.70}{\sqrt{12}}}\\ ={2.718}^{0.202073}\\ =1.2239\end{array}$

Calculate the value of the D,

$\text{D}=\frac{1}{\text{U}}$

Where,

• D is the Percentage of decrease in price of the house.
• U is the Percentage of increase in price of the house.

$\begin{array}{c}\text{D}=\frac{1}{1.2239}\\ =0.8170\end{array}$

Calculate the monthly risk free rate,

$\begin{array}{c}\text{Monthly}\text{risk}\text{free}\text{rate}=\frac{\text{Annual}\text{risk}\text{free}\text{rate}}{\text{12}}\\ =\frac{5\%}{12}\\ =0.42\%\end{array}$

Calculate the probability of price increase,

$\begin{array}{c}\text{Risk}\text{free}\text{rate}=\left({\text{P}}_{\text{Rise}}\right)\left({\text{Return}}_{\text{Fall}}\right)+\left({\text{P}}_{\text{Fall}}\right)\left({\text{Return}}_{\text{Fall}}\right)\\ 0.42\%=\left({\text{P}}_{\text{Rise}}\right)\left(22.39\right)\%+\left(1-{\text{P}}_{\text{Rise}}\right)\left(-18.3\%\right)\\ {\text{P}}_{\text{Rise}}=0.46\end{array}$

Calculate the probability of price decrease,

$\begin{array}{c}{\text{P}}_{\text{Fall}}=\left(1-{\text{P}}_{\text{Rise}}\right)\\ =1-0.46\\ =0.54\end{array}$

Calculate the stock price after one month in case, increase or decrease,

$\begin{array}{c}\text{Stock}\text{price}=\text{Current}\text{price}\times \text{Increase}\text{in}\text{stock}\text{price}\\ =\$83\times 1.2239\\ =\$101.58\end{array}$

Calculate the stock price when price decrease,

$\begin{array}{c}\text{Stock}\text{price}=\text{Current}\text{price}\times \text{Decrease}\text{in}\text{stock}\text{price}\\ =\$83\times 0.8170\\ =\$67.81\end{array}$

There will be four value of the stock in the two months, two values from the stock up and two values from the stock down,

Calculate the stock price after two month in case of stock up,

$\begin{array}{c}\text{Stock}\text{price}=\text{Current}\text{price}\times \text{Increase}\text{in}\text{stock}\text{price}\times \text{Increase}\text{in}\text{stock}\text{price}\\ =\$83\times 1.2239\times 1.2239\\ =\$124.32\end{array}$

Increase in the stock price is the 1.2239 for one month. Here, stock price is being to calculate two months that is why it will be multiplied two times.

Calculate the put option value in this case,

${\text{C}}_{\text{1}}=\text{max}\left(\text{S}-\text{X,0}\right)$

Where,

• Call value at the beginning is ${\text{C}}_0$.
• Strike price is X.
• House price at n period is S.

$\begin{array}{c}{\text{C}}_{\text{1}}=\text{Max}\left(\$90-\$124.32\right)\\ =-\$34.32\end{array}$

Calculate the value of the stock when price decrease,

$\begin{array}{c}\text{Stock}\text{price}=\text{Current}\text{price}\times \text{Decrease}\text{in}\text{stock}\text{price}\times \text{Increase}\text{in}\text{stock}\text{price}\\ =\$83\times 0.8170\times 1.2239\\ =\$82.99\end{array}$

Calculate the put option value in this case,

${\text{C}}_{\text{1}}=\text{max}\left(\text{S}-\text{X,0}\right)$

$\begin{array}{c}{\text{C}}_2=\text{Max}\left(\$90-\$82.99\right)\\ =\$7.01\end{array}$

Calculate again the stock price after one month in case, increase or decrease

$\begin{array}{c}\text{Stock}\text{price}=\text{Current}\text{price}\times \text{Increase}\text{in}\text{stock}\text{price}\times \text{Decrease}\text{in}\text{stock}\text{price}\\ =\$83\times 1.2239\times 0.8170\\ =\$82.99\end{array}$

Calculate the put option value in this case,

$C_0=\text{max}\left(\text{S}-\text{X,0}\right)$

$\begin{array}{c}{\text{C}}_2=\text{Max}\left(\$90-\$82.99\right)\\ =\$7.01\end{array}$

Calculate the value of the stock when price decrease,

$\begin{array}{c}\text{Stock}\text{price}=\text{Current}\text{price}\times \text{Decrease}\text{in}\text{stock}\text{price}\times \text{Decrease}\text{in}\text{stock}\text{price}\\ =\$83\times 0.8170\times 0.8170\\ =\$55.401\end{array}$

Decrease in the stock price is the 0.8170 for one month. Here, stock price is being to calculate two months that is why it will be multiplied two times.

Calculate the put option value in this case,

$C_0=\text{max}\left(\text{S}-\text{X,0}\right)$

$\begin{array}{c}{\text{C}}_2=\text{Max}\left(\$90-\$55.401\right)\\ =\$34.6\end{array}$

Conclusion

Hence, price of the put option is $1,377.1197.